I was at Barnes and Noble a few days ago and happened to be glancing through a film magazine called "Movie Insider"--I think. Anyways, there was a feature article about the cast of Lord of the Rings and their current projects. I read the brief article on Christopher Lee and found an interesting quote. I can't remember it word for word, but this quote is close enough.
Lee stated that Dooku is a very manipulative character. In fact, "as a disillusioned Jedi Master, I orchestrated the death of Liam Neeson in an episode I wasn't even in."
That makes sense given the fragmented timeline we were given in AOTC. Jango was recruited by "a man called Tyranus" ten years earlier. This coincides with the setting of TPM when Darth Maul was still alive. Sidious' purpose for Maul was most likely to assassinate Qui-Gon (I'll explain below).
During the deleted "Lost Twenty" segment, Jocasta Nu explains that Dooku left the order after Qui-Gon's death. However, it appears as though Dooku had already turned to the Dark Side and kept it concealed just like Palpatine. This would not only allow him to order the clones as Master Sifo-Dyas but also to have free access to the Jedi Archives where he could delete Kamino.
Dooku most likely reveals his role in Qui-Gon's murder during his duel with Anakin. Considering the close bond that Anakin had with Qui-Gon, that just might be enough to drive him temporarily into the Dark Side where he would find the strength to kill Count Dooku.
That might be why Palpatine ordered Maul to kill Qui-Gon. Depending on how far his vision into the future reaches, he may have foreseen it as a way to bring Anakin to his side. Also, it's a near parallel to when Vader taunts Luke with converting Leia to the Dark Side, and Luke attacks his father in an enraged state.